On dynamics of the mapping class group action on relative $\mathrm{PSL}(2,\mathbb{R})$-character varieties

Abstract

In this paper, we study the mapping class group action on the relative $\mathrm{PSL}(2,\mathbb{R})$-character varieties of punctured surfaces. It is well known that Minsky’s primitive-stable representations form a domain of discontinuity for the $\mathrm{Out}(F_n)$-action on the $\mathrm{PSL}(2,\mathbb{C})$-character variety. We define simple stability of representations of fundamental group of a surface into $\mathrm{PSL}(2,\mathbb{R})$ which is an analogue of the definition of primitive stability and prove that these representations form a domain of discontinuity for the $\mathrm{MCG}$-action. Our first main result shows that holonomies of hyperbolic cone surfaces are simple-stable. We also prove that holonomies of hyperbolic cone surfaces with exactly one cone-point of cone-angle less than $\pi$ are primitive-stable, thus giving examples of an infinite family of indiscrete primitive-stable representations.